Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.     

Calculating the knowledge-based similarity of functional groups using crystallographic data

A knowledge-based method for calculating the similarity of functional groups is described and validated. The method is based on experimental information derived from small molecule crystal structures. These data are used in the form of scatterplots that show the likelihood of a non-bonded interaction being formed between functional group A (the `central group') and functional group B (the `contact group' or `probe'). The scatterplots are converted into three-dimensional maps that show the propensity of the probe at different positions around the central group. Here we describe how to calculate the similarity of a pair of central groups based on these maps. The similarity method is validated using bioisosteric functional group pairs identified in the Bioster database and Relibase. The Bioster database is a critical compilation of thousands of bioisosteric molecule pairs, including drugs, enzyme inhibitors and agrochemicals. Relibase is an object-oriented database containing structural data about protein-ligand interactions. The distributions of the similarities of the bioisosteric functional group pairs are compared with similarities for all the possible pairs in IsoStar, and are found to be significantly different. Enrichment factors are also calculated showing the similarity method is statistically significantly better than random in predicting bioisosteric functional group pairs.     

Synthesis and thermal decomposition of potassium trioxalatoaluminate hydrate

The preparation of pure K₃Al(C₂O₄)₃·mH₂O (2<m<3) is described. Dependent on the mode of preparation, the following were found to be contaminants of the desired product: K₂C₂O₄·1H₂O; KHC₂O₄; KHC₂O₄·H₂C₂O₄·2H₂O; H₂C₂O₄·2H₂O; different forms of aluminium oxide hydrate; K₄Al₂(OH)₂(C₂O₄)₄· (2+x)H₂O (0.7<x<1.7) and K₂Al₂(H₂O)₂(C₂O₄)₄· 4H₂O.     

General order Newton-Padé approximants for multivariate functions

Summary Padé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off(x ₁, ...,x _p ). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value off at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that forp=1 the classical Newton-Padé approximants for a univariate function are obtained.     

Identification of human myocardial infarction in vitro based on the frequency dependence of ultrasonic backscatter.

It has been reported previously that acute and mature myocardial infarction in dogs can be differentiated in vitro and in vivo by ultrasonic tissue characterization based on measurement of the frequency dependence of ultrasonic backscatter. To characterize human infarction with an index of the frequency dependence of backscatter that could be obtained in patients, cylindrical biopsy specimens from 7 normal regions and 12 regions of infarction of 6 fixed, explanted human hearts in 2-deg steps around their entire circumference with a 5-MHz broadband transducer were insonified. One to six consecutive transmural levels were studied for each specimen. The dependence of apparent (uncompensated for attenuation or beam width) backscatter, /B(f)/2, on frequency (f) was computed from spectral analyses of radio-frequency data as /B(f)/2 = afn, where from theoretical considerations the magnitude of n decreases as scatterer size increases. Apparent integrated backscatter was computed as the average of /B(f)/2 from 3 to 7 MHz. The average value for n for normal tissue (0.9 +/- 0.1) exceeded that for tissue from regions of infarction (0.6 +/- 0.1; p less than 0.05). Infarct manifested a significant decrease of n from epicardial to endocardial levels (epi----mid----endo: 0.9----0.7----0.2; p less than 0.05) whereas normal tissue manifested similar values for n at each transmural level (0.8----1.1----0.9; p = NS). Average integrated backscatter across all transmural levels for infarct was significantly greater than for normal tissue (-48.3 +/- 0.5 vs -53.4 +/- 0.4 dB, infarct versus normal; p less than 0.05). The presence of fibrosis was associated with smaller values of n and greater integrated backscatter.(ABSTRACT TRUNCATED AT 250 WORDS)     

Anisotropy of ultrasonic velocity and elastic properties in normal human myocardium.

Measurements of ultrasonic quasilongitudinal velocity were made in the muscle fiber plane of excised human myocardium. Multiple adjacent planes across the left ventricular wall were interrogated to assess the transmural dependence of velocity. For each measurement plane, data were obtained in 2-deg increments through the full 360 deg relative to the myofibers. An approximate 1.3% magnitude of anisotropy was observed with maximum velocity along the muscle fibers and minimum velocity perpendicular to the muscle fibers. The known transmural shift in myofiber orientation was evidenced in the anisotropy of velocity as angular shifts between plots obtained from adjacent transmural planes within the same specimen. Measured values of velocity and density were used to estimate the effective C33 and C11 elastic constants of a thin layer of normal myocardium.